Fundamental polyhedra for all Deligne-Mostow lattices in PU(2,1)

TL;DR

This paper constructs fundamental domains for all Deligne-Mostow lattices in PU(2,1) with 2-fold symmetry, completing the classification in complex hyperbolic 2-space by extending previous symmetric cases.

Contribution

It introduces a method to build fundamental domains for Deligne-Mostow lattices with 2-fold symmetry, covering all commensurability classes in PU(2,1).

Findings

Constructed fundamental domains for all 2-fold symmetric Deligne-Mostow lattices.

Extended previous work from symmetric to asymmetric cone point configurations.

Complete classification of these lattices in complex hyperbolic 2-space.

Abstract

In this work we will build a fundamental domain for Deligne-Mostow lattices in PU(2,1) with 2-fold symmetry, which complete the whole list of Deligne-Mostow lattices in dimension 2. These lattices were introduced by Deligne and Mostow using monodromy of hypergeometric functions and have been reinterpreted by Thurston as authomorphisms on a sphere with cone singularities. Following his approach, Parker, Boadi and Parker, and Pasquinelli built a fundamental domain for the class of lattices with 3-fold symmetry, i.e. when three of five cone singularities have same cone angle. Here we extend this construction to the asymmetric case, where only two of the five cone points on the sphere have same cone angle, so to have a fundamental domain for each commensurability class of Deligne-Mostow lattices in PU(2,1).